Rider skill plays more than a "small" part. Left on their own, no horse would go over any of those jumps. They'd go around it, or just stop. And if the rider did nothing but apply leg and keep the horse straight, the horse might jump over, but the rider would certainly be off by no more than one stride after the landing.... Rider skill does play a small part, but the winner is invariably has the best horse. ...
It sounds like you're not a rider. I rode hunters over fences for over 20 years, up to 3'6", and competed for a few of those years, at amateur levels. It's common for non-riders to not understand the skill or the physical effort the rider puts in. The horse & rider are a team, and even a great horse will fail under a poor rider, while a mediocre horse under a great rider can do amazing things. At the Olympic level, you need both a great horse and a great rider.